# trig108's Question form Math Help Forum

#### Sudharaka

##### Well-known member
MHB Math Helper
Title: Website/Article of use for solving this?

trig108 said:
So we were given a couple practice problems to learn about characteristic polynomials..

However the book is very confusing to me and I can't find a whole lot of resource for two particular problems that are very similar...

If someone could point me in the right direction (material wise) to learning how to solve this I would appreciate it because I am very stumped.

USE the characteristic polynomial to solve the diffEq:

$2y''-2iy'+12y=0$

Please show me where I could learn to solve these types of problems, thank you!
Hi trig108, I think a good place to start with is Paul's Online Notes: Differential Equations. It gives a very simple introduction which is easy to understand. Also "A First Course in Differential Equations by Dennis Zill" is a nice book that I referred when doing some courses about differential equations. The Wikipedia article about Characteristic equations is another resource you might like.

Let me explain on how to solve the differential equation you have given,

$2y''-2iy'+12y=0$

Firstly we can divide the whole equation by $$2$$.

$\Rightarrow y''-iy'+6y=0$

Now we need a function that preserves its form after taking the first and second derivative. One such function is, $$y=e^{mx}$$ where $$m$$ is a constant to be determined (See this). When you substitute this into the differential equation,

$m^2e^{mx}-ime^{mx}+6e^{mx}=0$

Since, $$e^{mx}\neq 0$$ we get,

$m^2-im+6=0$

$m=\frac{i-\sqrt{23}}{2},\,m=\frac{i+\sqrt{23}}{2}$

Therefore the general solution of this differential equation would be,

$y(x)=A\,\mbox{exp}\,\left[\left(\frac{i-\sqrt{23}}{2}\right)x\right]+B\,\mbox{exp}\,\left[\left(\frac{i+\sqrt{23}}{2}\right)x\right]\mbox{ where }A\mbox{ and }B\mbox{ are arbitrary constants.}$

Kind Regards,
Sudharaka.

• Jameson